Chapter 9.1
ROTATIONAL MOTION
DESCRIBING ROTATIONAL MOTION
Eyad Omair
Rotational Quantities
To study the rotational motion, all linear (or translational)
quantities are replaced by rotational quantities.
linear displacement
linear velocity
linear acceleration
linear momentum
→ angular displacement
→ angular velocity
→ angular acceleration
→ angular momentum
Also the direction of vector quantities is denoted:
positive: counterclockwise (ccw)
negative: clockwise (cw)
ccw
cw
translational ≡ linear
rotational ≡ angular
Rotational Quantities
There are two units to measure the angle: degree and radian.
Angular Displacement (Δθ):
the change in the angle
unit: radian (rad)
∆𝜃 = 𝜃𝑓 − 𝜃𝑖
𝜋
𝜃 =
× 𝜃𝐷
180
𝑅
π = 3.14 rad
Revolution (rev) is another unit to measure the angular
displacement (1 rev = 2π rad).
Angular Velocity (ω):
the angular displacement divided by
the time required to make the rotation
unit: radian/second (rad/s)
∆𝜃
𝜔=
∆𝑡
There are two units to measure angular velocity:
rev/s = 2π rad/s
rev/min = rpm = 2π/60 rad/s
rev: revolution, round or cycle
Rotational Quantities
Angular Acceleration (α):
the change in angular velocity divided by the time required to make
that change.
unit: radian/second square (rad/s2)
∆𝜔
𝛼=
∆𝑡
Speeding up: angular velocity and angular acceleration have the same direction
Slowing down: angular velocity and angular acceleration have the different directions
Frequency (f):
the number of complete revolutions made by an object
in one second
unit: per second (s–1 or /s) Hertz (Hz)
ω is angular velocity and angular frequency
#
𝑓=
𝑡
𝜔
𝑓=
2π
Rotational Quantities
P.258
EX: Convert 180o from
degrees into radian
𝜋
𝑅
𝜃 =
× 𝜃𝐷
180
𝜃𝑅
P.237
𝜋
=
× 180
180
𝜃 𝑅 = 𝜋 = 3.14 rad
EX: Convert 2π from
radian into degrees
Rotational Quantities
P.237
7
P.237
P.258
Rotational Quantities
Angular velocity – time graph:
Slope represents angular acceleration
20 − 0
𝛼=
= 10 rad/s2
2−0
𝛼 = 0 rad/s2
t = 0 to t = 2
t = 2 to t = 4
Area represents angular displacement
1
∆𝜃 = 2 20 = 20 rad
2
∆𝜃 = 2 20 = 40 rad
t = 0 to t = 2
t = 2 to t = 4
Rotational Quantities
The relation between linear and angular quantities:
𝑥 = 𝑟𝜃
𝑎 = 𝑟𝛼
𝑣 = 𝑟𝜔
All points on the disc have:
Equal angular velocity
When an object rotates it may have
translational and rotational motion
at the same time e.g. bicycle
A
B
𝜔𝐴 = 𝜔𝐵
because they cover equal displacement (angle)
Different linear velocity 𝑣𝐴 > 𝑣𝐵
because its directly proportional to distance (r).
𝑟𝐴 > 𝑟𝐵
P.258
Rotational Quantities
When two gears are in contact and rotating. One is larger
than the other (different radii), or it can be imagined as a
truck tows a car.
Equal linear velocity
𝑣𝐴 = 𝑣𝐵
𝑎𝐴 = 𝑎𝐵
because they cover equal displacement
Different angular velocity
𝜔𝐴 < 𝜔𝐵
its inversely proportional to distance (r).
smaller gear rotates faster than big gear.
77
P.260
4
P.237
𝛼𝐴 < 𝛼𝐵
𝑟𝐴 > 𝑟𝐵
Rotational Quantities
P.258
P.237
Rotational Quantities
P.258
Ex: For the previous question, calculate the frequency